A Diagnostic Software Suite for Auditing Learned PDE Simulators

Abstract: Learned PDE simulators are increasingly used as low-cost replacements for expensive numerical solvers, but standard relative $L^2$ error does not determine whether a learned model behaves as a coherent numerical time propagator. This paper presents a diagnostic software suite for auditing learned PDE simulators as approximate evolution operators. The suite provides architecture-independent, post hoc diagnostics for relative state error, semigroup consistency, finite-difference generator discrepancy, energy behavior, integral balance, admissibility constraints, perturbation response, and scaling-law consistency. The software is designed around a minimal contract: reference trajectories, a learned propagator or saved predictions, equation metadata, and a diagnostic configuration specifying which structures are meaningful for the problem under study. We validate the suite on five benchmark PDE tasks: two-dimensional incompressible Navier-Stokes, shallow-water dynamics, active matter, three-dimensional compressible Navier-Stokes, and three-dimensional magnetohydrodynamics, using FNO, DeepONet, U-Net, and ResNet-style surrogate models together with controlled underfit and oversmoothed variants. The validation study shows that relative $L^2$ error can remain moderate, or even improve, while structural diagnostics deteriorate substantially. The package therefore supports software-level auditing of learned PDE simulators by reporting an interpretable diagnostic panel rather than collapsing model behavior into a single state-error score.